PRODUCT RULE FOR CAPUTO FRACTIONAL DERIVATIVES IS THE KEY TO MITTAG-LEFFLER AUTOCORRELATIONS IN VOLATILITY

The Caputo fractional derivative is the convolution of the normal derivative with a fractional kernel:

The fact that the derivative is in the integrand allows us to apply the usual product rule to

Now assume with a white noise . Since we have

where the white noise terms all have zero expectation. Then . Assuming the remainder term is small, this produces for the autocorrelation the fractional equation approximately therefore the autocorrelation will be a Mittag-Leffler function. This simple analysis is important because the empirical volatility autocorrelations are well-matched by and the analysis provides us with the steps toward modeling volatility with closer match to empirical data.

Now let us take a look at some of the fits to empirical data to get a sense for why these sorts of models are useful.